[discia] GCSE Maths › Geometry & Measures › Area and Circumference of Circles › Practise
  1. Either paper 1 mark
    Rosa adds labels to this diagram of a circle with centre \(O\). ABCDEFO
    LineRosa's label
    \(AB\)radius
    \(OC\)radius
    \(DE\)chord
    the line touching the circle at \(F\)tangent
    Explain why one of the labels is wrong.
    Show answer
    The line \(AB\) is labelled "radius" but it goes all the way across the circle through O; a radius only goes from the centre to the circumference.

    A radius joins the centre to a point on the circumference. A diameter joins two points on the circumference and passes through the centre. A chord joins two points on the circumference. A tangent touches the circle at exactly one point.

    Checking each line in the diagram against these: \(AB\) has both ends on the circle and passes through \(O\), so it is a diameter. \(OC\) runs from the centre \(O\) to the circle, so it is a radius. \(DE\) has both ends on the circle and misses \(O\), so it is a chord. The line at \(F\) meets the circle at one point only, so it is a tangent.

    Comparing these with the table, Rosa's label for \(AB\) is the one that does not match: it is a diameter, not a radius.

  2. Either paper 2 marks

    Here is a circular pond being made in a garden.

    O2 cm

    The radius of the circle is \(2\) cm.

    Work out the area of the circle. Give your answer in terms of \(\pi\).

    Show answer
    \(4\pi\ \text{cm}^2\)

    The area of a circle is \(\pi r^2\).

    \(A = \pi \times 2^2 = 4\pi\)

    Leaving the answer in terms of \(\pi\) means writing \(4\pi\) rather than multiplying it out, so nothing is lost to rounding.

    \(4\pi\) cm²

    • M1uses \(\pi r^2\) with \(r = 2\)
    • A1\(4\pi\) cm²
  3. Calculator 2 marks
    O11 cm

    The circle has a radius of \(11\) cm.

    Work out its area. Give your answer correct to 1 decimal place.

    Show answer
    \(380.1\ \text{cm}^2\)

    The area of a circle is \(\pi r^2\), so square the radius first and multiply by \(\pi\) afterwards.

    \(A = \pi \times 11^2\)

    Using the \(\pi\) key on a calculator gives \(380.1\) to 1 decimal place.

    \(380.1\) cm²

    • M1uses \(\pi \times 11^2\)
    • A1\(380.1\) cm²
  4. Either paper 2 marks
    ABO

    \(AB\) is a diameter of the circle and \(AB = 10\) cm.

    Calculate the circumference of the circle. Give your answer in terms of \(\pi\).

    Show answer
    \(10\pi\ \text{cm}\)

    The circumference is \(\pi d\), where \(d\) is the diameter — and the diameter is what the question has given, so no halving is needed.

    \(C = \pi \times 10 = 10\pi\)

    \(10\pi\) cm

    • M1uses \(\pi d\) with \(d = 10\)
    • A1\(10\pi\) cm
  5. Either paper 2 marks
    Or cm

    The circumference of the circle is \(14\pi\) cm.

    Find the radius of the circle.

    Show answer
    \(7\ \text{cm}\)

    The circumference is \(2\pi r\), so

    \(2\pi r = 14\pi\)

    Divide both sides by \(\pi\):

    \(2r = 14\)

    \(r = 7\)

    The radius is \(7\) cm.

    • M1forms \(2\pi r = 14\pi\) or \(2r = 14\)
    • A1radius \(7\) cm
  6. Either paper 3 marks
    (a)

    The diagram shows a circle with centre \(O\). \(AB\) is a diameter and \(AB = 22\) cm.

    ABO22 cm

    The shaded region is a semicircle.

    Calculate the perimeter of the shaded semicircle. Give your answer in terms of \(\pi\).

    (b)

    Calculate the area of the shaded semicircle. Give your answer in terms of \(\pi\).

    Show answer
    (a) \((11\pi + 22)\ \text{cm}\)
    (b) \(\frac{121}{2}\pi\ \text{cm}^2\)

    The perimeter of a semicircle is the curved part and the straight edge. Forgetting the straight edge is the mark most often dropped here.

    The curved part is half the circumference of the whole circle:

    \(\frac{1}{2} \times \pi \times 22 = 11\pi\)

    The straight edge is the diameter, \(22\) cm.

    \(P = 11\pi + 22\) cm

    Half the area of the whole circle:

    \(\frac{1}{2} \times \pi \times 11^2 = \frac{121}{2}\pi\)

    \(\frac{121}{2}\pi\) cm²

    • (a) M1curved part \(11\pi\) (half the circumference)
    • (a) A1adds the straight edge \(22\) cm: \(11\pi + 22\)
  7. Either paper 3 marks
    (a)

    \(OA\) and \(OB\) are radii of a circle with centre \(O\), and angle \(AOB = 90^\circ\).

    ABO6 cm90°

    \(OA = 6\) cm and the shaded region is a quarter of the circle.

    Work out the length of the arc \(AB\). Give your answer in terms of \(\pi\).

    (b)

    Work out the area of the shaded quarter circle. Give your answer in terms of \(\pi\).

    Show answer
    (a) \(3\pi\ \text{cm}\)
    (b) \(9\pi\ \text{cm}^2\)

    A right angle is a quarter of a full turn, so the arc is a quarter of the circumference.

    \(\frac{1}{4} \times 2 \times \pi \times 6 = 3\pi\)

    \(3\pi\) cm

    A quarter of the area of the whole circle:

    \(\frac{1}{4} \times \pi \times 6^2 = 9\pi\)

    \(9\pi\) cm²

    • (a) M1a quarter of the circumference, \(\frac{1}{4} \times 2 \times \pi \times 6\)
    • (a) A1arc length \(3\pi\) cm
  8. Calculator 3 marks
    (a)

    Priya's bicycle wheel has a diameter of \(50\) cm.

    O50 cm

    Calculate how far the wheel travels in one complete turn. Give your answer correct to 1 decimal place.

    (b)

    Calculate how far the wheel travels in \(40\) complete turns. Give your answer in metres, correct to 1 decimal place.

    Show answer
    (a) \(157.1\ \text{cm}\)
    (b) \(62.8\ \text{m}\)

    In one complete turn the wheel travels its own circumference.

    \(C = \pi d = \pi \times 50 = 157.1\) cm to 1 decimal place.

    Multiply one turn by \(40\), then divide by 100 to change centimetres into metres.

    \(157.1 \times 40 \div 100 = 62.8\) m

    • (a) M1uses \(\pi \times 50\) for the circumference
    • (a) A1distance \(157.1\) cm
  9. Calculator 3 marks

    Here is a circle with radius \(9\) cm.

    9 cm

    Here is a square with side \(14\) cm.

    14 cm

    The shapes are not drawn accurately.

    Which has the greater area, the circle or the square?

    You must show your working.

    Show answer
    The circle: circle \(\approx 254.5\ \text{cm}^2\), square \(196\ \text{cm}^2\)

    Area of the circle: \(\pi \times 9^2 = 81\pi \approx 254.5\) cm².

    Area of the square: \(14^2 = 196\) cm².

    Comparing \(254.5\) cm² with \(196\) cm².

    The circle has the greater area.

    • M1area of the circle \(\pi \times 9^2 \approx 254.5\) cm²
    • M1area of the square \(14^2 = 196\) cm²
    • A1The circle has the greater area, comparing 254.5 with 196
  10. Calculator 4 marks

    The plan of a patio is a rectangle with a semicircular pond cut out of one end. The pond is not paved.

    16 m16 m9 m

    The semicircle has the full width of the rectangle as its diameter.

    One pack of paving slabs covers \(116\) m².

    Is this enough to pave the whole patio? You must show your working.

    Show answer
    Yes: area \(\approx 112.19\) m² against \(116\) m²

    Rectangle: \(16 \times 9 = 144\) m².

    The diameter of the semicircle is \(9\) m, so its radius is \(4.5\) m.

    Semicircle: \(\frac{1}{2} \times \pi \times 4.5^2 \approx 31.81\) m².

    Subtract the semicircle from the rectangle: \(144 - 31.81 = 112.19\) m².

    The area is less than \(116\) m², so the answer is Yes.

    • M1rectangle area \(144\) m²
    • M1radius \(4.5\) m and semicircle area \(31.81\) m²
    • M1total area \(112.19\) m²
    • A1Yes, comparing \(112.19\) with \(116\)